Ray Matrix Approach for the Real TimeControl of SLR2000 Optical Elements

John J. Degnan

Sigma Space Corporation

14th International Workshop on Laser Ranging

San Fernando, Spain

7-11 June 2004

Basic 1D & 2D Ray Matrices*

d

1 d

0 1

Propagationover distance dx

Optic Axis

Thin Lensof FocalLength, f

1 0

-1/f 1

Optic Axis

Mirror Surfacewith Curvature R

1 0

-2/R 1

Optic Axis

Dielectric Interface

Optic Axis

n21 0

0 n1/n2n1

ax

I

dII

0

1D 2D

IIf

I1

0

-

IIR

I2

0

-

In

nI

2

10

0

a

x

10

01=I

y

x

y

x

a

a

*Valid only for a linear optical system. We need to perform a coordinatetransformation whenever the beam changes direction

Example of Coordinate SystemChange: Canted Mirror

mirror

Optical axis

Optical axisOff-axis

Ray

api+1

api’

S-vector into page

di

di+1

pi+1

pi'

( )

s

p

s

p

iii

ii

i

iiii

x

x

sdp

sdp

ddss

a

a

q

1000

0100

0010

0001

ˆˆˆ

'ˆˆ'ˆ

2sin

ˆˆ'ˆˆ

111

1

11

-

-

¥=

¥=

¥==

+++

++

Simplified SLR2000 Transceiver*

Automated Devices•Star CCD cameraperiodically updates mountmodel•3x telescope compensates forthermal drift in main telescopefocus•Beam magnifier controlslaser spot size and divergenceat exit aperture•Risley prism pair controlstransmitter point-ahead•Variable iris controls receiverfield of view (FOV)•Quadrant detector providesfine pointing corrections

*Planned modifications in red

CCDCamera

0.4 m

0.088 m5.4XBeam

Reducer

0.04 m

0.025 m

CCDSplitter

QUADMirror Transmitter

CompensatorBlock

TelescopePit Mirror

FaradayIsolator

0.080 m

0.070 m

Polarizers

Polarizer

0.085 m 0.105 m 0.350 m

0.070 m

“Zero”Wedge

Day/NightFilters

3-elementTelephoto

Lensf=0. 85 m

0.130 m

LaserTransmitter

0.213 m

Risley Prisms

Computer-controlled

Diaphragm/Spatial Filter

(Min. Range : 0.5to 6 mm diameter)

QuadrantRangingDetector

0.200 m0.051 m

3 X Telescope

f1 = -0.1 m

0.099 m

NORTH(a = 0o)

a0 = 67.4o

f2 =0.3 m

d1

d2

d3(out of page)

f1 = 0.108 m

f2 = -0.02 m

f = .025 m

d1

p1

d1

p1

d1

p1

d2

p2

Vector out of page

Vector into page

p3

s3

d2

s2'

0.250 mWorkingDistance

0.258 m

SpecialOptics2x-8xBeam

Expander

Translation Stage

Transceiver Bench Matrices

01

10 -=L

L

LL=

333.00

267.231aMTransmitter

Quadrant Detector

Star Camera0469.2

405.0559.351 L-

LL-=cM

General Form

LL

LL=

xx

xxx dc

baM1

L-L-

LL=

101.0392.0

57.2079.01bM

Coude Mount Matrix

C1

C2

C3Telescope

TurningMirror

FromTelescopePit Mirror

AzimuthAxis

ElevationAxis

Coude TrackingMount (not to scale)

s-Vector out of page

s -Vector into page

d4

p4

d5

p5

d6

p6

d5

p5'

d4

p4'

d4

d5

d6To Telescope

d3

d3

p3'

0.64 m

0.792 m

0.31 m

InstantaneousAzimuth

out of page

gg

gg

cossin

sincos

-

--=G

eaag --= 0

G

GG=

02Cd

M

a= mount azimuth anglee = mount elevation anglea0 = azimuth angle of transceiver axis at the Coude pit mirror = 67.4o (SLR2000)dC = Coude path length = 1.742 m

Telescope Assembly Matrix

1.2819 m

1.2192 m

0.4 m

Rp = 3.048 m

Rs = - .8128 m

SecondaryMirror

Primary Mirror

TelescopeTurning Mirror

Window

ft = - 0.6 m0.6214 m 0.1905 m

From CoudeTrackingMount

SLR2000TELESCOPE

(NOT TO SCALE)

0.211 m

9.97o

s-Vector out of page

s -Vector into page

d9

p9

d8

p8

d8

p8'

Elevation Axis

d7

p7d7

p7'

d9

+ Az

+ El s9

Im

IdImM

T

TT

103 = mT = telescope magnification = 10.16

dT = 5.758 m

Total SLR2000 System Matrix

( )( )

t

xx

t

xx

xTxCxtx

xTxCxtx

xx

xxxx

m

dD

m

cC

ddddbmB

cdcdamA

DC

BAMMMM

=

=

++=

++=

--

-=G

GG

GG==

gg

gg

sincos

cossin'

''

''123

Outgoing Rays Incoming Rays

gg

gg

sincos

cossin'

''

''1

-

--=G

GG-

G-G=-

T

Tx

Tx

Tx

Tx

xAC

BDM

x = a Transmitter b Quadrant Detector c Star Camera

LL

LL=

xx

xxx dc

baM1

Star Calibrations

p-axis

s-axis

Optical Bench

Looking throughrear of Star Camera

p-axis

s-axis

Looking throughtelescope window

s sc

psc

as9

Mount Elevation Axis

To Ground

ap9

CCD array

s

p

n

n

pixel

arcgg

gege

e

a

sincos

cossecsinsecsec5.0 -=

D

D

Da = star azimuth offsetDe = star elevation offsetnp = CCD pixel columnns = CCD pixel row

eaag --= 0

Quadrant Pointing Correction

p-axis

s-axis

Optical Bench

Looking throughrear of quadrant PMT

p-axis

s-axis

Looking throughtelescope window

sqd

pqd

ap9

as9

Mount Elevation Axis

To Ground

Satellite

Q1Q2

Q3 Q4

c

c

s

p

mm

arcgg

gege

e

a

sincos

cossecsinsecsec5.10 -=

D

D

Da = azimuth pointing correctionDe = elevation pointing correctionpc = horizontal centroid coordinatesc = vertical centroid coordinate

eaag --= 0

Receiver Field of View

Da = iris diameterFOV = Full Receiver Field of View

in arcsec

FOVarc

mmDa sec

125.0=

Da

TTaTaa

TT

a

T

T

TT

T

a

a

arc

mm

rad

mxxx

rad

mx

xx

aa

a

aa

sec

125.0908.25

'908.25

'387.24'039.0

'908.250

===

G-=

G-G

G-=

rr

rr

r

r

r

r

Stepper-controlled Iris

Transmitter Point-Ahead

Optical Bench

p-axis

s-axis

Looking throughtelescope window

as9

Mount Elevation Axis

To Ground

ap9

“Apparent”Position

(receiver axis)

Future Position(transmitter

axis)

ActualSatellitePosition

p-axis

s-axis

Wedge 1

Wedge 2

Looking through RisleyPrisms toward

telescope

x1x2

Dxe

a

geg

gegt

a

a

&

&

sincoscos

coscossin

-

--= rT

rp

rp ms

p

aprp =Risley prism output angleprojected into p planeasrp = Risley prism output angleprojected into s planemT = post-Risley magnification of transmitter =30.48tr = pulse roundtrip time of flight

•

•

e

a = azimuth rate

= elevation rate

eaag --= 0

Computing Risley Prism Orientations

r

rT

rp

rp ms

p

te

ta

geg

geg

xdxd

xdxda

a

&

&

sincoscos

coscossin

sinsin

coscos

2211

2211

-

--=

+

+=

d1 = half cone angle traced by wedge 1d2 = half cone angle traced by wedge 2x1 = wedge 1 angle relative to home positionx2 = wedge 2 angle relative to home position

( ) ( ) ( )( ) ( )( )[ ]( )xdddd

xgdgdtexgdgdetax

D++

D-++D-+-=

cos2

cos(cossinsincoscos

2122

21

21211

rrTm &&

( ) ( )( ) ( ) ( )( )[ ]( )xdddd

xgdgdtexgdgdetax

D++

D-+-D-+=

cos2

sinsincoscoscos.)sin(

2122

21

21211

rrTm &&

( )[ ] ( )˛˝¸

ÓÌÏ +-+

=-≡D -

21

22

21

2221

12 2

)cos(cos

ddddteeta

xxx rrTm &&

Solve above two equations for two unknown Risley orientations x1 and x2:

x2 = x1 + Dx

Simulated LAGEOS Pass

440 460 480 500 520174.5

175

175.5Differential Wedge Angle vs Time

Dxii

deg

PCA

tii

min

440 460 480 500 520180

90

0

90

180

270

360Risley Orientations vs Time

x1ii

deg

x2ii

deg

PCA

tii

mina( ) b( )

440 460 480 500 5200.001

5 .104

0

5 .104

0.001Differential Angular Rate vs Time

Time (min)

Differential Angular Rate (deg/sec)

PCA

440 460 480 500 5200.05

0.025

0

0.025

0.05Risley Angular Rates vs Time

Time (min)

Wedge Angular Rates (deg/sec)

PCA

c( ) d( )

Azimuth- Elevation Offsets & Beam Centering

40 20 0 20 4040

20

0

20

40Transmitter Point-Ahead

Azimuth Offset (arcsec)

Elevation Offset (arcsec)

4 2 0 2 44

2

0

2

4Central Ray Trajectory on Exit Window

p-axis (increasing azimuth), cm

s-axis (increasing elevation), cm

a( ) b( )

Ray Matrices and Gaussian Beams

Paraxial ray matrix theory can be applied to gaussian beampropagation if we define the following complex parameter:

( ) ( )zj

zRzq 2)(

11

pwl

-=

If propagation from a point z0 to z can be described by the ray matrix

DC

BAM =

then the gaussian beam properties at z are given by

( )

( )0

0

1

1

)(

1

zqBA

zqDC

zq +

+=

Controlling Beam Divergence

The ray matrix which takes the transmitter beam from theRisley Prism output to the far field is of the form

'1

0

''

'1

0

''

0lim

G

GG=G

GG=

•Æ

t

tt

t

tt

r

m

m

rm

m

dm

I

rIIFF

where mt is the total transmitter magnification. From our gaussian parameter, we obtain the following for the full beam divergence

( )( )

( )( )

( )( )

2

0

02

min

2

0

02

0

112

2 ˜̃¯

ˆÁÁË

Ê+=˜̃

¯

ˆÁÁË

Ê+==

zR

z

zR

z

zmr

r

tt l

pwq

lpw

wplw

q

where w(z0) and R(z0) are the beam radius and phasefront radius of curvature out of the computer-controlled telescope in the transmit path. To first order, beam divergence varies linearly withthe lens displacement from perfect focus.

Beam Divergence vs Phase FrontCurvature at Risleys

Radius of SLR2000 primary, a = 20 cmOptimum spot radius at window*,wopt = a/1.12 = 17.9 cmPost-Risley magnification, mt = 30.48Optimum beam radius at Risley,w(zo) = wopt /mt = 5.9 mmMinimum Divergence,qmin = 0.388 arcsec

0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

Inverse Phase Front Curvature, m^-1

Full Beam Divergence, arcsec

q jj q min 1p w0

2⋅

lRinv

jj⋅

ÊÁÁË

ˆ̃

˜¯

2

+⋅:=

Summary• Ray matrix approach provides us with the

mathematical tools to calculate in real time:– Scale factor and angular rotation for converting star

image offsets from center in the CCD camera toazimuth and elevation biases

– Scale factor and angular rotation for convertingquadrant centroid position to satellite pointingcorrection in az-el space

– Transmitter point ahead as a function of round triptime-of-flight and the instantaneous azimuthal andelevation angular rates

– Iris diameter (spatial filter) setting for a given receiverFOV

– Transmitter beam size divergence as a function oftransmit telescope defocus